PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

432 Practical MATLAB® Applications for Engineers

Example 4.19

The switch in the circuit diagram of Figure 4.77 closes at t = 0, where it remains for t > 0.

The storing energy elements such as the (two) capacitors and the (two) inductors are

initially charged as indicated in Figure 4.77 (for t = 0 ).

1. Redraw the equivalent circuit in the s-domain


2. Obtain in the s-domain the loop equations


3. Obtain the matrix loop equation of part 2


4. Use MATLAB and obtain expressions for I 1 (s) and I 2 (s)


5. Obtain the system’s poles


6. Use MATLAB and verify the initial and fi nal value loop currents by applying the
   initial and fi nal value theorems


7. Use MATLAB and obtain plots of I 1 (s) versus s and I 2 (s) versus s


8. Discuss the results

The MATLAB solution is given as follows by e script fi le IC_loop_diff_eqs.

ANALYTICAL Solution

Part 1

The time domain circuit shown in Figure 4.77 is transformed into the equivalent

s-domain circuit shown in Figure 4.78, where the initial conditions are converted into

sources, and the assumed loop currents directions of I 1 (s) and I 2 (s) are indicated.

Part 2

The two loop equations of the circuit diagram of Figure 4.78 are

For loop #1:

50

2

3

5

22

123

ss

s

s

Is

s


 Is















() ()

For loop #2:

3

9

415

9

2

337

5

sss ^2  
 sIs^1 s s I

















() 22 ()s

v 1 (t) = 50V

V 2

sw closes at t = 0 L 1 = 1 H R 1 L 2 = 3 H

= 2 Ω

R 2 = 3 Ω

R 3 = 4 Ω

C 1 = 0.5 F

C 2 = 0.33 F

IL1(o) = 2 A

IL2(0) = 3 A

+

VC1(0) = 3 V

−

+

VC2(0) = 4 V

−

(t) = 5sin(3t) V

+

−

FIGURE 4.77
Circuit diagram of Example 4.19.